tissues like organs, muscle, and skin. This allows us to use X-ray photographs
to differentiate body tissues.) We can define a scattering factor fAfor atom Aas
a way to describe the ability of individual Aatoms to scatter X rays. The higher
the scattering factor, the better an atom scatters X rays, and so the stronger the
final diffracted X ray. As you might expect, in general larger atoms have higher
scattering factors than smaller atoms. Figure 21.25 shows diffraction patterns
for NaCl and KCl—two very similar compounds but with ions of different sizes
and, therefore, scattering factors. Although the diffractions show a similar scat-
tering pattern, the intensities of similar diffractions are obviously different.
The intensity of a particular X-ray diffraction also depends partly on phase
effects. Recall that electromagnetic radiation can constructively interfere or de-
structively interfere, as shown in Figure 21.16. For some crystals in which a dif-
fraction from a plane of atoms is expected to occur, it turns out that diffrac-
tion from an adjacent plane contributes X rays of the exact opposite phase:the
result is complete destructive interference, and the intensity of this expected
diffraction is zero. This is the reason that odd values of the sum h^2 k^2 ^2
are absent for body-centered unit cells (see Table 21.3). For crystals that are
composed of different atoms having similar scattering factors, there may be ac-
cidental destructive interferences that can dramatically reduce the intensity of
an expected diffraction.
We finish this section on Miller indices by introducing a convenient use
of Miller indices. We like to define a solid crystal as an infinite, regular array
of atoms or molecules. In reality, however, we know that the array is not in-
finite; the crystal stops at some point. It stops at the surface of the crystal. In
many cases, the surfaceof a crystal is not just some random arrangement of
atoms or molecules making a microscopically rough boundary. For many
crystals, over a large surface area (that is, on a scale of square nanometers or
micrometers) the surface corresponds to a particular plane of atoms or mol-
ecules that can be described by a particular set of Miller indices. Figure 21.2621.6 Miller Indices 751NaCl KCl0 °180 °
Figure 21.25 NaCl and KCl have the same
unit cell, but the different sizes of the Naand
Kions cause a different intensity of some indi-
vidual diffractions.
Figure 21.26 Surfaces of crystals can also
be described using Miller indices. In fact,
the cutting and polishing of many gemstones
follows specific Miller index planes.
Mineralogists, gemologists, and lapidaries
must know these planes in order to properly
cut and polish gemstones.
© Charles D. Winters/Photo Researchers, Inc.© Sinclair Strammers/Science Photo Library/PhotoResearchers, Inc. © Charles D. Winters/Photo Researchers, Inc.